# Properties

 Label 104.6.f.a Level $104$ Weight $6$ Character orbit 104.f Analytic conductor $16.680$ Analytic rank $0$ Dimension $18$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [104,6,Mod(25,104)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("104.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$104 = 2^{3} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 104.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$16.6799172605$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} + 2988 x^{16} + 3538582 x^{14} + 2154924500 x^{12} + 739476805665 x^{10} + 146816139850952 x^{8} + \cdots + 16\!\cdots\!04$$ x^18 + 2988*x^16 + 3538582*x^14 + 2154924500*x^12 + 739476805665*x^10 + 146816139850952*x^8 + 16446222036643216*x^6 + 945928045851032064*x^4 + 22287212188303003648*x^2 + 160915244038046613504 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{63}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + \beta_{8} q^{5} - \beta_{10} q^{7} + (2 \beta_{2} - \beta_1 + 90) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + b8 * q^5 - b10 * q^7 + (2*b2 - b1 + 90) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + \beta_{8} q^{5} - \beta_{10} q^{7} + (2 \beta_{2} - \beta_1 + 90) q^{9} - \beta_{12} q^{11} + ( - \beta_{15} + \beta_{10} + \cdots + 19) q^{13}+ \cdots + (40 \beta_{17} - 20 \beta_{16} + \cdots + 939 \beta_{8}) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + b8 * q^5 - b10 * q^7 + (2*b2 - b1 + 90) * q^9 - b12 * q^11 + (-b15 + b10 + b8 - 2*b2 + 19) * q^13 + (b16 - b11 + b10) * q^15 + (-b3 - 17*b2 - 173) * q^17 + (-3*b11 + b9 + b8) * q^19 + (-b17 + b12 + 3*b11 + 3*b10 - 4*b8) * q^21 + (b5 + b3 + 12*b2 + 2*b1 - 353) * q^23 + (b15 - b14 + b5 + b4 - b3 + 8*b2 - 845) * q^25 + (-3*b15 + 3*b14 - b7 - b4 - 2*b3 - 138*b2 + 4*b1 - 465) * q^27 + (2*b15 - 2*b14 - b6 + b5 - b4 - 50*b2 - b1 + 77) * q^29 + (-2*b17 + b16 + 2*b15 + 2*b14 + b13 + 3*b11 + 7*b10 + b9 - 3*b8) * q^31 + (-3*b17 + 4*b16 - b15 - b14 + b13 + 2*b12 + 13*b11 + 12*b10 - 2*b9 - 15*b8) * q^33 + (-5*b15 + 5*b14 - b7 + b6 + b5 + b4 - 2*b3 - 60*b2 - 6*b1 - 560) * q^35 + (b17 + 4*b16 - 5*b15 - 5*b14 - 2*b13 + b12 - 20*b11 + 5*b10 + 4*b9 + 24*b8) * q^37 + (2*b17 + 3*b16 - b15 + b14 + b13 - 8*b12 - 18*b11 - 8*b10 + 3*b9 + 3*b8 + b7 - b6 + b4 - 3*b3 + 31*b2 - 12*b1 + 624) * q^39 + (-b17 - 4*b16 - 5*b15 - 5*b14 - b13 - 10*b12 - 2*b11 - 12*b10 + 2*b9 + 11*b8) * q^41 + (-4*b15 + 4*b14 + 2*b7 + b6 + b5 - 4*b3 + 55*b2 + 2*b1 - 840) * q^43 + (5*b17 + 4*b16 - 10*b15 - 10*b14 - 25*b12 - 30*b11 + 29*b10 + 4*b9 + 75*b8) * q^45 + (-6*b17 + 6*b16 + 10*b15 + 10*b14 - 3*b13 + 4*b12 + 28*b11 - 40*b10 - 7*b9 - 7*b8) * q^47 + (7*b15 - 7*b14 + 2*b6 - b5 - b4 - 10*b3 - 151*b2 - 5*b1 - 1237) * q^49 + (-13*b15 + 13*b14 - b7 - b6 - b5 + b4 + 6*b3 + 84*b2 - 46*b1 + 5666) * q^51 + (7*b15 - 7*b14 - 4*b7 + b6 + b5 - b4 + 10*b3 - 46*b2 - 29*b1 + 3317) * q^53 + (-6*b15 + 6*b14 + 2*b7 - 6*b6 - b5 - 2*b4 + 5*b3 - 256*b2 + 42*b1 - 697) * q^55 + (-10*b15 - 10*b14 + 10*b13 + 30*b12 + 33*b11 - 34*b10 - 8*b9 - 14*b8) * q^57 + (-4*b17 - 16*b16 + 20*b15 + 20*b14 + 8*b13 + 31*b12 + 54*b11 - 46*b10 - 2*b9 + 58*b8) * q^59 + (9*b15 - 9*b14 - 4*b7 + 5*b6 - b5 + b4 + 106*b2 + 43*b1 - 5011) * q^61 + (6*b17 - 15*b16 + 10*b15 + 10*b14 - 17*b13 - 62*b12 - 5*b11 - 21*b10 - b9 - 121*b8) * q^63 + (3*b17 - 12*b16 + 10*b15 - 26*b14 + 9*b13 + 28*b12 - 4*b11 + 80*b10 + 2*b9 + 137*b8 + 4*b7 - 6*b6 + b5 + b4 + 10*b3 + 95*b2 + 35*b1 - 1678) * q^65 + (4*b17 + 10*b16 + 20*b15 + 20*b14 - 8*b13 + 34*b12 - 47*b11 + 94*b10 - b9 - 61*b8) * q^67 + (31*b15 - 31*b14 + 8*b7 + 7*b6 - b5 + b4 + 4*b3 + 1008*b2 + 37*b1 - 3681) * q^69 + (8*b17 - 10*b16 + 12*b15 + 12*b14 + 2*b13 + 60*b12 - 2*b11 - 143*b10 - 252*b8) * q^71 + (b17 - 20*b16 - 15*b15 - 15*b14 - 23*b13 + 2*b12 - 7*b11 + 140*b10 - 6*b9 - 313*b8) * q^73 + (-31*b15 + 31*b14 + b7 + 11*b6 - 11*b5 - b4 - 4*b3 + 597*b2 + 66*b1 - 1981) * q^75 + (14*b15 - 14*b14 - 4*b7 - 7*b6 - 11*b5 - 9*b4 + 6*b3 - 822*b2 + 121*b1 - 3495) * q^77 + (-10*b15 + 10*b14 - 2*b7 + 4*b6 + 10*b4 + 16*b3 + 684*b2 + 92*b1 - 9370) * q^79 + (27*b15 - 27*b14 + 8*b7 - 12*b6 - 11*b5 + 9*b4 + 9*b3 + 1512*b2 - 215*b1 + 23772) * q^81 + (12*b17 - 6*b16 + 28*b15 + 28*b14 + 42*b13 - 12*b12 + 25*b11 + 182*b10 - 7*b9 + 209*b8) * q^83 + (16*b17 + 4*b16 - 21*b15 - 21*b14 + 28*b13 - 4*b12 + 41*b11 + 102*b10 - 265*b8) * q^85 + (-6*b15 + 6*b14 - 2*b7 + 6*b6 - 11*b5 - 10*b4 + 19*b3 - 532*b2 - 106*b1 + 16613) * q^87 + (-10*b15 - 10*b14 - 60*b13 + 60*b12 + 63*b11 - 152*b10 - 20*b9 + 230*b8) * q^89 + (-8*b17 - 2*b16 + 10*b15 + 30*b14 + 34*b13 + 5*b12 + 62*b11 - 188*b10 - 16*b9 - 396*b8 - 11*b6 - b5 - 10*b4 + 30*b3 + 545*b2 - 70*b1 + 17914) * q^91 + (11*b17 - 20*b16 - 11*b15 - 11*b14 - 26*b13 - 101*b12 - 82*b11 - 461*b10 - 16*b9 + 63*b8) * q^93 + (18*b6 - 10*b5 - 28*b3 - 366*b2 + 40*b1 - 5740) * q^95 + (-29*b17 - 20*b16 + 15*b15 + 15*b14 + 13*b13 - 24*b12 + 243*b11 + 342*b10 - 26*b9 + 643*b8) * q^97 + (40*b17 - 20*b16 - 70*b13 - 130*b12 - 367*b11 - 588*b10 + 39*b9 + 939*b8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q - 18 q^{3} + 1620 q^{9}+O(q^{10})$$ 18 * q - 18 * q^3 + 1620 * q^9 $$18 q - 18 q^{3} + 1620 q^{9} + 348 q^{13} - 3122 q^{17} - 6348 q^{23} - 15232 q^{25} - 8358 q^{27} + 1360 q^{29} - 10046 q^{35} + 11228 q^{39} - 15090 q^{43} - 22428 q^{49} + 102186 q^{51} + 59668 q^{53} - 12416 q^{55} - 90336 q^{61} - 30310 q^{65} - 66532 q^{69} - 35288 q^{75} - 63040 q^{77} - 168428 q^{79} + 427730 q^{81} + 299264 q^{87} + 322814 q^{91} - 103524 q^{95}+O(q^{100})$$ 18 * q - 18 * q^3 + 1620 * q^9 + 348 * q^13 - 3122 * q^17 - 6348 * q^23 - 15232 * q^25 - 8358 * q^27 + 1360 * q^29 - 10046 * q^35 + 11228 * q^39 - 15090 * q^43 - 22428 * q^49 + 102186 * q^51 + 59668 * q^53 - 12416 * q^55 - 90336 * q^61 - 30310 * q^65 - 66532 * q^69 - 35288 * q^75 - 63040 * q^77 - 168428 * q^79 + 427730 * q^81 + 299264 * q^87 + 322814 * q^91 - 103524 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 2988 x^{16} + 3538582 x^{14} + 2154924500 x^{12} + 739476805665 x^{10} + 146816139850952 x^{8} + \cdots + 16\!\cdots\!04$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + 332$$ v^2 + 332 $$\beta_{2}$$ $$=$$ $$( 18\!\cdots\!29 \nu^{16} + \cdots + 16\!\cdots\!44 ) / 19\!\cdots\!04$$ (1807665145664938429*v^16 + 5081725048660429245932*v^14 + 5498480417797801413041150*v^12 + 2924220315603461907278066340*v^10 + 820857613750189537013563924125*v^8 + 120938015456343639441526571490584*v^6 + 8542750528085860665634121368842320*v^4 + 224606520605241728829816737484744448*v^2 + 1636262756720016265032035717987794944) / 19476223380890087724265871703803904 $$\beta_{3}$$ $$=$$ $$( 16\!\cdots\!79 \nu^{16} + \cdots + 18\!\cdots\!92 ) / 57\!\cdots\!64$$ (164294359759904344795808279*v^16 + 445741503181925767673022997069*v^14 + 457855561761020760565032602499733*v^12 + 225676614358538034785402151304340367*v^10 + 57112270441123880034557401932771512544*v^8 + 7450565975207927543711608945314820298032*v^6 + 484244397937349410007106097474588513274752*v^4 + 15352643843700923662532487971845013507753984*v^2 + 183651827517973579283218137311599364412014592) / 57193889565690066004979296855157746827264 $$\beta_{4}$$ $$=$$ $$( - 25\!\cdots\!93 \nu^{16} + \cdots - 16\!\cdots\!72 ) / 74\!\cdots\!32$$ (-2566918701045779187452426593*v^16 - 7897680792992562330478816769735*v^14 - 9631866523208788808344414944456599*v^12 - 5986307701283877549860477977245076557*v^10 - 2038427847386833857190274408616884732172*v^8 - 379964113942146583287194182585842905973296*v^6 - 36496239044660489808425992907506453299388736*v^4 - 1528899488333001020216362739052869869874086912*v^2 - 16370188245823966909037534966578258559962447872) / 743520564353970858064730859117050708754432 $$\beta_{5}$$ $$=$$ $$( 21\!\cdots\!99 \nu^{16} + \cdots + 17\!\cdots\!44 ) / 24\!\cdots\!44$$ (2167596732538558550937989399*v^16 + 6019291118696848547592736318193*v^14 + 6409832259745370718272675481493121*v^12 + 3345024087297842144471896339499178139*v^10 + 923957521227163505338012570205470950404*v^8 + 136157382458752751458987517906728534094288*v^6 + 9998489965713768549675243577488012390429632*v^4 + 291513537016330145471011482075473205744749568*v^2 + 1762245666196516098281053596173217530318290944) / 247840188117990286021576953039016902918144 $$\beta_{6}$$ $$=$$ $$( - 48\!\cdots\!15 \nu^{16} + \cdots - 68\!\cdots\!08 ) / 24\!\cdots\!44$$ (-4868365920083383084368506515*v^16 - 13594820458984544780043274486197*v^14 - 14582289847478074109732961099649989*v^12 - 7675647412828275478817180843736536071*v^10 - 2135845057175028457966569071571908799668*v^8 - 314893111112030451830196494956162408383504*v^6 - 22851232885620699870011112514969885759638720*v^4 - 672929271625871749312105392720129150575025152*v^2 - 6853029323247597173185240929196381812203716608) / 247840188117990286021576953039016902918144 $$\beta_{7}$$ $$=$$ $$( - 17\!\cdots\!03 \nu^{16} + \cdots - 87\!\cdots\!24 ) / 37\!\cdots\!16$$ (-17750975023909723779280620103*v^16 - 49403681717091244956190358563559*v^14 - 52673934228881953242728201619668675*v^12 - 27407663817985947689196341641736075565*v^10 - 7452653928820579459748857480673592178986*v^8 - 1047421220627755824715142145722093504532800*v^6 - 68310012720297283972063412799025791951431456*v^4 - 1497358871264140063688941564067168195233169920*v^2 - 8743796352879430173084063617123013436691841024) / 371760282176985429032365429558525354377216 $$\beta_{8}$$ $$=$$ $$( - 30\!\cdots\!83 \nu^{17} + \cdots + 39\!\cdots\!36 \nu ) / 65\!\cdots\!44$$ (-306437867096204558943043291120483*v^17 - 757688830484273577153975019868570636*v^15 - 653464306727264055437835590284832439146*v^13 - 213773585341765614263971800614597609588292*v^11 - 3229398109011844899993963761752613453826363*v^9 + 12807423587982977445426284260839016350167213464*v^7 + 2671583922996615914563531482511219312002654672976*v^5 + 194235265066558365592697549679888035289564315268352*v^3 + 3905179703184572198216930021523992701756945419534336*v) / 65498158038867862398814202008101750295809170079744 $$\beta_{9}$$ $$=$$ $$( 17\!\cdots\!63 \nu^{17} + \cdots + 21\!\cdots\!88 \nu ) / 19\!\cdots\!32$$ (1713310272671316393881228792178763*v^17 + 14061451975529271463206836476947431648*v^15 + 30361823799337815688845808628933329251254*v^13 + 28654199203142111710305496422231217901599584*v^11 + 13500350108251918389395928909601244065203249687*v^9 + 3269858079171757402143886468699901547410679291464*v^7 + 389240386955201536402384774307901718205143815381872*v^5 + 19196105878029744087223968198946781579211483533477632*v^3 + 213237381101691988298438001362278603782383826854412288*v) / 196494474116603587196442606024305250887427510239232 $$\beta_{10}$$ $$=$$ $$( - 38\!\cdots\!87 \nu^{17} + \cdots - 30\!\cdots\!60 \nu ) / 17\!\cdots\!12$$ (-381899557694870748701172316715587*v^17 - 1093995650257149958235300379533901992*v^15 - 1224269668586740383408842823689476238078*v^13 - 693393387860294619652643764967008965675768*v^11 - 219110193737656957308527157369958481403986103*v^9 - 40120085795496376170061808848107149132819862728*v^7 - 4155850242000588087935878205217251562891087759728*v^5 - 210140145049604664129623532428180862156389934688000*v^3 - 3003970650239457294058348085383706583817757637672960*v) / 17863134010600326108767509638573204626129773658112 $$\beta_{11}$$ $$=$$ $$( 13\!\cdots\!43 \nu^{17} + \cdots + 26\!\cdots\!40 \nu ) / 20\!\cdots\!84$$ (1343640909713437529926943*v^17 + 3775937983664374019763607732*v^15 + 4083095122248908851793054743210*v^13 + 2168887125926615422696379462712300*v^11 + 607190991325172553240728135567042175*v^9 + 88801780280278608943369652012599251736*v^7 + 6117322797351869363339317218318654922096*v^5 + 142166607558584658562519596030729900680192*v^3 + 266978795862736824568893867084693324042240*v) / 20105836123179966812650145604840698755584 $$\beta_{12}$$ $$=$$ $$( 85\!\cdots\!91 \nu^{17} + \cdots + 37\!\cdots\!60 \nu ) / 54\!\cdots\!12$$ (859301678247360878478123290980791*v^17 + 2459872037088662909683816477074810167*v^15 + 2738585613002602968540818761551381882003*v^13 + 1525696470626472697008817540444327909140125*v^11 + 462443116648741802280156925765245017510312618*v^9 + 77587564142507944693327697091765894036270676032*v^7 + 6918694536241473978865275407053267768441427150624*v^5 + 287632258842319591194050574378923054223429105517056*v^3 + 3700987064017729490310582813930290469686449288642560*v) / 5458179836572321866567850167341812524650764173312 $$\beta_{13}$$ $$=$$ $$( - 31\!\cdots\!53 \nu^{17} + \cdots - 11\!\cdots\!28 \nu ) / 19\!\cdots\!32$$ (-31461043985625373384639392684066953*v^17 - 90426442085445193459496751733619694652*v^15 - 101153388168662970001895861355671620249414*v^13 - 56613378531149108026291584794618054725996500*v^11 - 17185404537718817261093527144610249423894356137*v^9 - 2862606110304600838037407730196682377693531071992*v^7 - 249280472975639457535994716952731451144001897360016*v^5 - 9903328927790692385775770043081827148953482545633536*v^3 - 112351881827767695552535192911260059466385499519254528*v) / 196494474116603587196442606024305250887427510239232 $$\beta_{14}$$ $$=$$ $$( 37\!\cdots\!09 \nu^{17} + \cdots - 73\!\cdots\!60 ) / 19\!\cdots\!32$$ (37563593652114603418769900657555409*v^17 - 138425102981078901829785380268499168*v^16 + 105409555818461240048893560206511115524*v^15 - 393231695093308096713433012661001325472*v^14 + 113787777807023245899085385647473803660334*v^13 - 431514848633737451433635771876771426784416*v^12 + 60342986107469829816591421163055188105559660*v^11 - 233564507801332073393186466698001452119181280*v^10 + 16889824541001661297548354040110749383899694809*v^9 - 66656519017172449359844122301871139830930322432*v^8 + 2482902076830723500489188549199032554870470254584*v^7 - 9813666749969318378740968600367806355513433245184*v^6 + 175625125881279859169023693997641423628358888139152*v^5 - 647673846211332000046092936650746570144111846092800*v^4 + 4874530575804467880612861599211061344890730691168512*v^3 - 12154916159566217661457009876809106601127157835038720*v^2 + 67228142188757086457640871232151357076400806045483008*v - 73879453723031861227870891961986076038237911578050560) / 196494474116603587196442606024305250887427510239232 $$\beta_{15}$$ $$=$$ $$( 37\!\cdots\!09 \nu^{17} + \cdots + 73\!\cdots\!60 ) / 19\!\cdots\!32$$ (37563593652114603418769900657555409*v^17 + 138425102981078901829785380268499168*v^16 + 105409555818461240048893560206511115524*v^15 + 393231695093308096713433012661001325472*v^14 + 113787777807023245899085385647473803660334*v^13 + 431514848633737451433635771876771426784416*v^12 + 60342986107469829816591421163055188105559660*v^11 + 233564507801332073393186466698001452119181280*v^10 + 16889824541001661297548354040110749383899694809*v^9 + 66656519017172449359844122301871139830930322432*v^8 + 2482902076830723500489188549199032554870470254584*v^7 + 9813666749969318378740968600367806355513433245184*v^6 + 175625125881279859169023693997641423628358888139152*v^5 + 647673846211332000046092936650746570144111846092800*v^4 + 4874530575804467880612861599211061344890730691168512*v^3 + 12154916159566217661457009876809106601127157835038720*v^2 + 67228142188757086457640871232151357076400806045483008*v + 73879453723031861227870891961986076038237911578050560) / 196494474116603587196442606024305250887427510239232 $$\beta_{16}$$ $$=$$ $$( 41\!\cdots\!55 \nu^{17} + \cdots + 56\!\cdots\!60 \nu ) / 98\!\cdots\!16$$ (41410356807801367323138351633075055*v^17 + 115365740320655169054830197902837966242*v^15 + 123342754331081805926000593041009628772420*v^13 + 64644767495159455840979400264493456745114550*v^11 + 17904061693461884648108626875952650173895967053*v^9 + 2633275888794694586648137247420918500448323564664*v^7 + 192505773984114094589477597625876556029627451011536*v^5 + 5852962152956711533193811801527848710825180497685760*v^3 + 56993262633320339488596898137262424718977091835330560*v) / 98247237058301793598221303012152625443713755119616 $$\beta_{17}$$ $$=$$ $$( 13\!\cdots\!31 \nu^{17} + \cdots + 36\!\cdots\!44 \nu ) / 98\!\cdots\!16$$ (139071727744831292918768003199080531*v^17 + 386780091252011492053029538409907916888*v^15 + 411896464352854395496646234886924273741358*v^13 + 213882767392794955796828045999518384906920840*v^11 + 57945053500706979064835419194504136891135141047*v^9 + 8086989680552446378097525949609329429274910210504*v^7 + 519353397211458206470806060479713013128430708683120*v^5 + 10737048684599352006054013714951795087281112213776128*v^3 + 36124190787724710719490140094920016027908936976236544*v) / 98247237058301793598221303012152625443713755119616
 $$\nu$$ $$=$$ $$( 2\beta_{13} + 2\beta_{12} + \beta_{11} + 2\beta_{10} + 4\beta_{8} ) / 128$$ (2*b13 + 2*b12 + b11 + 2*b10 + 4*b8) / 128 $$\nu^{2}$$ $$=$$ $$\beta _1 - 332$$ b1 - 332 $$\nu^{3}$$ $$=$$ $$( 16 \beta_{17} - 384 \beta_{16} + 368 \beta_{15} + 368 \beta_{14} - 970 \beta_{13} + \cdots - 2868 \beta_{8} ) / 128$$ (16*b17 - 384*b16 + 368*b15 + 368*b14 - 970*b13 - 1642*b12 + 31*b11 - 4026*b10 + 16*b9 - 2868*b8) / 128 $$\nu^{4}$$ $$=$$ $$15 \beta_{15} - 15 \beta_{14} + 4 \beta_{7} - 12 \beta_{6} - 11 \beta_{5} + 5 \beta_{4} + \beta_{3} + \cdots + 205671$$ 15*b15 - 15*b14 + 4*b7 - 12*b6 - 11*b5 + 5*b4 + b3 + 482*b2 - 934*b1 + 205671 $$\nu^{5}$$ $$=$$ $$( - 52688 \beta_{17} + 471168 \beta_{16} - 334512 \beta_{15} - 334512 \beta_{14} + \cdots + 2709780 \beta_{8} ) / 128$$ (-52688*b17 + 471168*b16 - 334512*b15 - 334512*b14 + 666874*b13 + 1314330*b12 + 9273*b11 + 3924778*b10 - 33744*b9 + 2709780*b8) / 128 $$\nu^{6}$$ $$=$$ $$- 12549 \beta_{15} + 12549 \beta_{14} - 5510 \beta_{7} + 14586 \beta_{6} + 12225 \beta_{5} + \cdots - 158029869$$ -12549*b15 + 12549*b14 - 5510*b7 + 14586*b6 + 12225*b5 - 7463*b4 + 2267*b3 - 1056892*b2 + 817662*b1 - 158029869 $$\nu^{7}$$ $$=$$ $$( 69711952 \beta_{17} - 459159936 \beta_{16} + 284980528 \beta_{15} + 284980528 \beta_{14} + \cdots - 2421549556 \beta_{8} ) / 128$$ (69711952*b17 - 459159936*b16 + 284980528*b15 + 284980528*b14 - 529707626*b13 - 1092998794*b12 - 193919249*b11 - 3480979482*b10 + 41962832*b9 - 2421549556*b8) / 128 $$\nu^{8}$$ $$=$$ $$10113063 \beta_{15} - 10113063 \beta_{14} + 6011200 \beta_{7} - 14111640 \beta_{6} - 11348539 \beta_{5} + \cdots + 131198809319$$ 10113063*b15 - 10113063*b14 + 6011200*b7 - 14111640*b6 - 11348539*b5 + 8197101*b4 - 2944635*b3 + 1417797254*b2 - 711175794*b1 + 131198809319 $$\nu^{9}$$ $$=$$ $$( - 75785848272 \beta_{17} + 416813398656 \beta_{16} - 241248793776 \beta_{15} + \cdots + 2068777878484 \beta_{8} ) / 128$$ (-75785848272*b17 + 416813398656*b16 - 241248793776*b15 - 241248793776*b14 + 444239935322*b13 + 940693831802*b12 + 326482088233*b11 + 3034431484298*b10 - 46149828048*b9 + 2068777878484*b8) / 128 $$\nu^{10}$$ $$=$$ $$- 9456013449 \beta_{15} + 9456013449 \beta_{14} - 6090102330 \beta_{7} + 12765524646 \beta_{6} + \cdots - 112080821503313$$ -9456013449*b15 + 9456013449*b14 - 6090102330*b7 + 12765524646*b6 + 10045613445*b5 - 8171731155*b4 + 2212777755*b3 - 1606884715248*b2 + 618915280774*b1 - 112080821503313 $$\nu^{11}$$ $$=$$ $$( 76590796870480 \beta_{17} - 368165022706560 \beta_{16} + 204010351060016 \beta_{15} + \cdots - 17\!\cdots\!24 \beta_{8} ) / 128$$ (76590796870480*b17 - 368165022706560*b16 + 204010351060016*b15 + 204010351060016*b14 - 380719553380042*b13 - 826119246601450*b12 - 404400032707457*b11 - 2640297042533754*b10 + 47821801295440*b9 - 1714958746796724*b8) / 128 $$\nu^{12}$$ $$=$$ $$9732341661963 \beta_{15} - 9732341661963 \beta_{14} + 5975664113428 \beta_{7} - 11266228366548 \beta_{6} + \cdots + 96\!\cdots\!19$$ 9732341661963*b15 - 9732341661963*b14 + 5975664113428*b7 - 11266228366548*b6 - 8725809517391*b5 + 7835469114809*b4 - 1069053985019*b3 + 1688855839615706*b2 - 539340189993946*b1 + 96872473306692219 $$\nu^{13}$$ $$=$$ $$( - 74\!\cdots\!28 \beta_{17} + \cdots + 13\!\cdots\!12 \beta_{8} ) / 128$$ (-74847171650667728*b17 + 321508226848894080*b16 - 172438161277386672*b15 - 172438161277386672*b14 + 329439251652972730*b13 + 733082027348888538*b12 + 445910056368114201*b11 + 2299631609612914666*b10 - 47880023190590160*b9 + 1393068905506533012*b8) / 128 $$\nu^{14}$$ $$=$$ $$- 10\!\cdots\!13 \beta_{15} + \cdots - 84\!\cdots\!89$$ -10254857599340013*b15 + 10254857599340013*b14 - 5762579748683726*b7 + 9845757893792034*b6 + 7512876496964313*b5 - 7378049967453791*b4 - 50496999034021*b3 - 1705970232299073220*b2 + 470563940133923886*b1 - 84183260341338388389 $$\nu^{15}$$ $$=$$ $$( 71\!\cdots\!28 \beta_{17} + \cdots - 11\!\cdots\!44 \beta_{8} ) / 128$$ (71828877263559592528*b17 - 279432605156041817472*b16 + 145715699880719931184*b15 + 145715699880719931184*b14 - 286436463410237593898*b13 - 653814853587530291530*b12 - 464463948458508954353*b11 - 2005457428339952272602*b10 + 46918473820591553360*b9 - 1114865637873988829044*b8) / 128 $$\nu^{16}$$ $$=$$ $$10\!\cdots\!23 \beta_{15} + \cdots + 73\!\cdots\!59$$ 10698258192402856623*b15 - 10698258192402856623*b14 + 5495173096276905448*b7 - 8570916329807904528*b6 - 6441585272501665699*b5 + 6880247713438213701*b4 + 994958294235798981*b3 + 1682793108219873141038*b2 - 410987488319924187522*b1 + 73368685043805131668559 $$\nu^{17}$$ $$=$$ $$( - 68\!\cdots\!84 \beta_{17} + \cdots + 88\!\cdots\!40 \beta_{8} ) / 128$$ (-68161131000947560645584*b17 + 242431645788256075996800*b16 - 123125261354581357955760*b15 - 123125261354581357955760*b14 + 249717490525582312043546*b13 + 584495622194136149287226*b12 + 468217716676483838356489*b11 + 1750895589892479130024010*b10 - 45312042146628389234640*b9 + 881118818466404312755540*b8) / 128

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/104\mathbb{Z}\right)^\times$$.

 $$n$$ $$41$$ $$53$$ $$79$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 29.8951i − 29.8951i − 15.8775i 15.8775i 14.6049i − 14.6049i − 11.7289i 11.7289i − 3.57287i 3.57287i − 5.41828i 5.41828i 13.4499i − 13.4499i 20.6516i − 20.6516i − 29.0138i 29.0138i
0 −30.8951 0 38.1208i 0 141.460i 0 711.509 0
25.2 0 −30.8951 0 38.1208i 0 141.460i 0 711.509 0
25.3 0 −16.8775 0 109.079i 0 10.3732i 0 41.8507 0
25.4 0 −16.8775 0 109.079i 0 10.3732i 0 41.8507 0
25.5 0 −15.6049 0 44.2438i 0 223.936i 0 0.514183 0
25.6 0 −15.6049 0 44.2438i 0 223.936i 0 0.514183 0
25.7 0 −12.7289 0 2.61467i 0 44.8479i 0 −80.9747 0
25.8 0 −12.7289 0 2.61467i 0 44.8479i 0 −80.9747 0
25.9 0 2.57287 0 100.064i 0 184.172i 0 −236.380 0
25.10 0 2.57287 0 100.064i 0 184.172i 0 −236.380 0
25.11 0 4.41828 0 15.1584i 0 136.551i 0 −223.479 0
25.12 0 4.41828 0 15.1584i 0 136.551i 0 −223.479 0
25.13 0 12.4499 0 24.0434i 0 155.838i 0 −87.9990 0
25.14 0 12.4499 0 24.0434i 0 155.838i 0 −87.9990 0
25.15 0 19.6516 0 66.6761i 0 50.5832i 0 143.186 0
25.16 0 19.6516 0 66.6761i 0 50.5832i 0 143.186 0
25.17 0 28.0138 0 71.8258i 0 103.875i 0 541.773 0
25.18 0 28.0138 0 71.8258i 0 103.875i 0 541.773 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.6.f.a 18
3.b odd 2 1 936.6.c.c 18
4.b odd 2 1 208.6.f.e 18
13.b even 2 1 inner 104.6.f.a 18
39.d odd 2 1 936.6.c.c 18
52.b odd 2 1 208.6.f.e 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.6.f.a 18 1.a even 1 1 trivial
104.6.f.a 18 13.b even 2 1 inner
208.6.f.e 18 4.b odd 2 1
208.6.f.e 18 52.b odd 2 1
936.6.c.c 18 3.b odd 2 1
936.6.c.c 18 39.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(104, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$(T^{9} + 9 T^{8} + \cdots - 8069766912)^{2}$$
$5$ $$T^{18} + \cdots + 70\!\cdots\!96$$
$7$ $$T^{18} + \cdots + 92\!\cdots\!96$$
$11$ $$T^{18} + \cdots + 13\!\cdots\!24$$
$13$ $$T^{18} + \cdots + 13\!\cdots\!93$$
$17$ $$(T^{9} + \cdots + 22\!\cdots\!76)^{2}$$
$19$ $$T^{18} + \cdots + 28\!\cdots\!00$$
$23$ $$(T^{9} + \cdots + 23\!\cdots\!44)^{2}$$
$29$ $$(T^{9} + \cdots - 49\!\cdots\!72)^{2}$$
$31$ $$T^{18} + \cdots + 11\!\cdots\!24$$
$37$ $$T^{18} + \cdots + 87\!\cdots\!76$$
$41$ $$T^{18} + \cdots + 59\!\cdots\!00$$
$43$ $$(T^{9} + \cdots + 41\!\cdots\!52)^{2}$$
$47$ $$T^{18} + \cdots + 72\!\cdots\!84$$
$53$ $$(T^{9} + \cdots - 12\!\cdots\!60)^{2}$$
$59$ $$T^{18} + \cdots + 57\!\cdots\!00$$
$61$ $$(T^{9} + \cdots - 29\!\cdots\!24)^{2}$$
$67$ $$T^{18} + \cdots + 14\!\cdots\!24$$
$71$ $$T^{18} + \cdots + 38\!\cdots\!24$$
$73$ $$T^{18} + \cdots + 15\!\cdots\!56$$
$79$ $$(T^{9} + \cdots - 29\!\cdots\!76)^{2}$$
$83$ $$T^{18} + \cdots + 50\!\cdots\!24$$
$89$ $$T^{18} + \cdots + 25\!\cdots\!56$$
$97$ $$T^{18} + \cdots + 34\!\cdots\!00$$